It has been proven that a quantum computer is superior to an electronic com puter in solving some NP problems. Based on quantum operations, this article proposes a new quantum sum and quantum multiplication, and then the floating point quantum multiplier and quantum sum are created on the basis of fixed point number operations. These studies lay the foundation for the quantum implementation of digital filters. This article provides a new way to calculate the summator on a quantum computer. This method uses the quantum Fourier transform (QFT) and reduces the number of qubits needed to be added by eliminating the need to temporarily transfer bits. This approach also allows you to add a classical number to a quantum superposition without encoding a classical number in a quantum register. This method also allows for mass parallelization during its execution. Adding and multiplying capabilities based on QFT are improved with some changes. The proposed operations are compared with the operations of close quantum arithmetic.
This paper proposes a new and improved implementation of a quantum integer multiplier. Performing arithmetic computations is sometimes a necessary step in the implementation of quantum algorithms. In this work, Quantum Fourier Transform is used in order to perform scalable arithmetic in a generic bit-width quantum system. In the phase domain, addition can be implemented through accumulated controlled rotations on the qubits’ state. Leveraging this, and inspired by the classical implementation of an array multiplier, a new integer multiplier is fully designed and tested in a quantum environment.The depth of a quantum circuit is the number of computational steps necessary to completion, and it is a key parameter that reflects on the performance of the design. The new design reduces the quantum depth of the design from the exponential order of the previously proposed designs to polynomial order.
We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) . In this approach the DFrFT is defined as the
N×N dimensional unitary representation of the generator of the arithmetic rotational group
SO2[ZN], which is the finite set of
modN integer,
2×2 matrices acting on the points of the discrete toroidal phase space lattice
ZN×ZN, preserving the euclidean distance
modN. We construct explicitly, using techniques of the Finite Quantum Mechanics (FQM), the
pn dimensional unitary matrix representation of the group
SO2[Zpn] and especially we work out in detail the one which corresponds to the generator. This is our definition of the arithmetic fractional Fourier transform (AFrFT). Following this definition, we proceed to the construction of efficient quantum circuits for the AFrFT, on sets of
n p-dimensional qudits with
p a prime integer, by introducing novel quantum subcircuits for diagonal operators with quadratic phases as well as new quantum subcircuits for multipliers by a constant. The quantum subcircuits that we introduce provide a set capable to construct quantum circuits for any element of a more general group, the group of Linear Canonical Transformations (LCT),
SL2[ZN] of the toroidal phase space lattice. As a byproduct, extensions of the diagonal and multiplier quantum circuits for both the qudit and qubit case are given, which are useful alone in various applications. Also, we analyze the depth, width and gate complexity of the efficient AFrFT quantum circuit and we estimate its gate complexity which is of the order
O(n2), its depth which is of the order
O(n) with depth
n, while at the same time it has a structure permitting local interactions between the qudits.